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Find sum of all possible values of θ in the interval (-π/2, π) for which [latex]frac{3+2isin theta }{1-2isin theta}[/latex] is purely imaginary;

a) π/3b) πc) 2π/3d) π/2Solution: sin2 θ = 3/4As θ∈ (-π/2,π ) = ±π/3, 2π/3Sum of all values of θ = 2π/3Answer: (c)... View Article

There is a parabola having axis as x-axis, vertex is at a distance of 2 unit from origin & focus is at (4, 0). Which of the following point does not lie on the parabola.

a) (6, 8)b) (5, 2√6)c) (8, 4√3)d) (4,-4)Solution:The equation of parabola is y2 = 8(x-2)(6,8) does not lie on this curve.Answer: (a)... View Article

[latex]\lim_{y\to 0}\frac{\sqrt{1+\sqrt{1+y^{4}}}-\sqrt{2}}{y^{4}}[/latex] =

a) 1/4√2b) 1/2√2c) 1/2√2(1+√2)d) does not existSolution:Rationalising numerator,Answer: (a)... View Article

{2⁴⁰³/15} = k/15 then find k. (where {.} denotes fractional part function)

a) 2b) 8c) 1d) 4Solution:Solution:24 ≡ 1 (mod 15)2400 ≡ 1 (mod 15)2403 ≡ 8(mod 15){2403/15} = 8/15k = 15Answer: (b)... View Article

a,b,c are in G.P. a+b+c = bx , then x cannot be;

a) 2b) -2c) 3d) 4Solution:Let the terms of G.P be a/r, a, ar(a/r) + a+ar = axx = r+(1/r)+1But r+(1/r) ≥ 2Or r+(1/r) ≤ -2 ( using A.M,G.M... View Article

Average height and variance of 5 students in a class is 150 and 18 respectively. A new student whose height is 156 cm is added to the group. Find new variance.

a) 20b) 22c) 16d) 14Solution:Let 5 students are x1 , x2 , x3 , x4 , x5Given = [(x12 + x22+ x32+ x42+ x52+ x62)/6]-(151)2From equation (2) and... View Article

Check if function is continuous or discontinuous

Question: If then:a) f (x) is discontinuous ∀ a ∈ R, b ∈ Rb) f (x) is continuous if a = 0 & b = 5c) f (x) is continuous if... View Article

If [latex]\vec{a}=i -j[/latex] , [latex]\vec{b}=i +j+k[/latex] are two vectors, and c is another vector such that [latex]\vec{a}\times \vec{c}+\vec{b}=\vec{0}[/latex] and [latex]\vec{a}. \vec{c}=4[/latex] then [latex]\left | \vec{c}\right |^{2} =[/latex]

a) 9b) 8c) 19/2d) 17/2Solution: Answer: (c)... View Article

If x² ≠ nπ+1, n ∈N, then [latex]\int x \sqrt{\frac{2 \sin \left(x^{2}-1\right)-\sin 2\left(x^{2}-1\right)}{2 \sin \left(x^{2}-1\right)+\sin 2\left(x^{2}-1\right)}} d x[/latex] is equal to

a) Answer: (c)... View Article

[latex]\int_{0}^{\pi }\left | \cos x \right |^{3}dx[/latex] is equal to;

a) 4/3b) 2/3c) 0d) 8/3Solution:Answer: (a)... View Article

The equation x²+2x+2 = 0 has roots α and β. Then value of α¹⁵ + β¹⁵is

a) 512b) 256c) -512d) -256Solution:Solution: (x+1)2 = -1 ⇒ x+1 = ±ix = -1+I, -1-i α = -1+I, β = -1-i α = √2ei(3π/4), β =... View Article

There are 5 girls and 7 boys. A team of 3 boys and 2 girls is to be formed such that no two specific boys are in the same team. Number of way to do so

a) 400b) 250c) 200d) 300Solution:Total number of teams = 7C3× 5C2= 35×10 = 350Let A, B be the specific boys.Number of teams with... View Article

The sum [latex]sum _{k=1}^{20}(1+2+3+ ……+k)[/latex] is

Solution: =(1/2)[2870+210]=1540... View Article

The least positive value of ′ ′ for which the equation, 2 ²+( −10) +(33/2) =2 has real roots is

Solution: ∵2x2+(a−10)x+33/2=2a, a ∈ 𝐙+ has real roots =>D ≥ 0 =>(a−10)2−4×2×((33/2)−2a) ≥ 0 =>(a−10)2−4(33−4a) ≥ 0 =>a2−4a−32 ≥... View Article

Let the normal at a on the curve ² − 3 ² + + 10 = 0 intersect the y-axis at (0,3/2). If is the slope of the tangent at to the curve, then | | is equal to

Let co-ordinate of P be (x1, y1) Differentiating the curve w.r.t x 2yy′ − 6x + y′ = 0 Slope of tangent at P => y′ =6x1/(1 + 2y1) mnormal=((... View Article

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at most three of them are red is

Solution: Number of ways to select at most 3 red balls = P(0 red balls) +P(1 red ball) + (2 red balls ) + P(3 red balls) = 7 4 + 5 1 × 7 3 + 5... View Article

Let f(x) = x cos⁻¹ (sin(−|x|)) , x ∈ (− /2 , /2 ), then which of the following is true ?

a. f ′ (0) = − π/2b. f ′ is decreasing in (− π/2 , 0) and increasing in (0, π/2 )c. f is not differentiable at x = 0d. f ′ is increasing in (−... View Article

If C is a point at which Rolle’s theorem holds for the function, f(x) = [latex]log_{e}\left ( \frac{x^{2} + \alpha }{7x} \right )[/latex] in the interval [3, 4], where α ∈ 𝐑, then f′′(c) is equal to :

a. −1/24b. −1/12c. √3/7d. 1/12Solution:Rolle’s theorem is applicable on f(x) in [3, 4]⇒ f(3) = f(4)⇒ l ⇒9 + α/21 =16 + α/28⇒ 36 + 4α = 48 + 3α ⇒... View Article

Let the line y = mx and the ellipse 2x² + y² = 1 intersect a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at (−1/3√2,0) and (0, ), then is equal to

a. 2/√3b. 2/3c. 2√2/3d. √2/3Solution: Let ≡ (x1, y1)2x2 + y2 = 1 is given equation of ellipse.⇒ 4x + 2yy’=0⇒ y′|(x1,y1) = −2x1/y1 Therefore,... View Article

Let two points be A(1, −1) and B(0, 2). If a point P(x′, ′) be such that the area of ΔPAB = 5 sq. units and it lies on the line, 3x + − 4 = 0, then the value of lambda is

a. 4b. 1c. −3d. 3Solution:Area of triangle is 3x’+y’=12 or 3x’+y’=-8⇒ = 3 or −2... View Article